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The Hyperbolic Schur Decomposition Šego, Vedran 2012 The hyperbolic Schur decomposition Šego, Vedran 2012 MIMS EPrint: 2012.119 Manchester Institute for Mathematical Sciences School of Mathematics The University of Manchester Reports available from: http://eprints.maths.manchester.ac.uk/ And by contacting: The MIMS Secretary School of Mathematics The University of Manchester Manchester, M13 9PL, UK ISSN 1749-9097 The hyperbolic Schur decomposition Vedran Segoˇ a,b a Faculty of Science, University of Zagreb, Croatia b School of Mathematics, The University of Manchester, Manchester, M13 9PL, UK Abstract We propose a hyperbolic counterpart of the Schur decomposition, with the em- phasis on the preservation of structures related to some given hyperbolic scalar product. We give results regarding the existence of such a decomposition and research the properties of its block triangular factor for various structured ma- trices. Keywords: indefinite scalar products, hyperbolic scalar products, Schur decomposition, Jordan decomposition, quasitriangularization, quasidiagonalization, structured matrices 2000 MSC: 15A63, 46C20, 65F25 1. Introduction The Schur decomposition A = UTU ∗, sometimes also called Schur’s unitary triangularization, is a unitary similarity between any given square matrix A ∈ n n n n C × and some upper triangular matrix T ∈ C × . Such a decomposition has a structured form for various structured matrices, i.e., T is diagonal if and only if A is normal, real diagonal if and only if A is Hermitian, positive (nonnegative) real diagonal if and only if A is positive (semi)definite and so on. Furthermore, the Schur decomposition can be computed in a numerically stable way, making it a good choice for calculating the eigenvalues of A (which are the diagonal elements of T ) as well as the various matrix functions (for more details, see [11]). Its structure preserving property allows to save time and memory when working with structured matrices. For example, computing the value of some function of a Hermitian matrix is reduced to working with a diagonal matrix, which involves only evaluation of the diagonal elements. Unitary matrices are very useful when working with the traditional Euclidean n scalar product hx,yi = y∗x, as their columns form an orthonormal basis of C . However, many applications require a nonstandard scalar product which is usu- ally defined by [x,y]J = y∗Jx, where J is some nonsingular matrix, and many of these applications consider Hermitian or skew-Hermitian J. The hyperbolic Email address: [email protected] (Vedran Sego)ˇ scalar product defined by a signature matrix J = diag(j1,...,jn)(jk ∈ {−1, 1}) arises frequently in applications. It is used, for example, in the theory of rel- ativity and in the research of the polarized light. More on the applications of such products can be found in [10, 13, 14, 17]. The Euclidean matrix decompositions have some nice structure preserving properties even in nonstandard scalar products, as shown by Mackey, Mackey and Tisseur [16], but it is often worth looking into versions of such decomposi- tions that respect the structures related with the given scalar product. There is plenty of research on the subject, i.e., hyperbolic SVD [17, 24], J1J2-SVD [9], two-sided hyperbolic SVD [20], hyperbolic CS decomposition [8, 10] and indefinite QR factorization [19]. There are many advantages of using decompositions related to some specific, nonstandard scalar product, as such decompositions preserve structures related to a given scalar product. They can simplify calculation and provide a better insight into the structures of such structured matrices. In this paper we investigate the existence of a decomposition which would resemble the traditional Schur decomposition, but with respect to the given hyperbolic scalar product. In other words, our similarity matrix should be unitary-like (orthonormal, to be more precise) with respect to that scalar prod- uct. As we shall see, a hyperbolic Schur decomposition can be constructed, but not for all square matrices. Furthermore, we will have to relax conditions on both U and T . The matrix U will be hyperexchange (a column-permutation of the matrix unitary with respect to J). The matrix T will have to be block upper triangular with diagonal blocks of order 1 and 2. Both of these changes are quite usual in hyperbolic scalar products. For example, they appear in the traditional QR vs. the hyperbolic QR factorizations [19]. Some work on the hyperbolic Schur decomposition was done by Ammar, Mehl and Mehrmann [1, Theorem 8], but with somewhat different focus. They have assumed to have a partitioned J = Ip ⊕ (−Iq), in the paper denoted as Σp,q, for which they have observed a Schur-like similarity through unitary factors (without permuting J), producing more complex triangular factors. Also, their decomposition is applicable only to the set of J-unitary matrices, in the paper denoted as the Lie group Op,q. In the symplectic scalar product spaces, Schur-like decomposition was re- searched by Lin, Mehrmann and Xu [15], by Ammar, Mehl and Mehrmann [1], and by Xu [22, 23]. In section 2, we provide a brief overview of the definitions, properties and other results relating to the hyperbolic scalar products that will be used later. In section 3, the definition and the construction of the hyperbolic Schur decom- position are presented. We also provide sufficient requirements for its existence and examples showing why such a decomposition does not exist for all matrices. In section 4 we observe various properties of the proposed decomposition. We finalize the results by providing the necessary and the sufficient conditions for the existence of the hyperbolic Schur decomposition of J-Hermitian matrices in section 5. 2 The notation used is fairly standard. The capital letters refer to matrices and their blocks, elements are denoted by the appropriate lowercase letter with two subscript indices, while lowercase letters with a single subscript index represent vectors (including matrix columns). By J = diag(±1) we denote a diagonal signature matrix defining the hyperbolic scalar product, while P and Pk (for some indices k) denote permutation matrices. We use 1 . 1 Sn := [δi,n+1 j ]= .. = Sn− − 1 for the standard involutory permutation (see [6, Example 2.1.1.]), J for a Jor- dan matrix and Jk(λ) for a single Jordan block of order k associated with the th eigenvalue λ. Vector ek denotes k column of the identity matrix and ⊗ denotes the Kronecker product. The symbol ⊕ is used to describe a diagonal concatena- tion of matrices, i.e., A ⊕ B is a block diagonal matrix with the diagonal blocks A and B. Also a standard notation, but somewhat incorrect in terms of the indefinite scalar products, is |v| := |[v,v]|. This is used as the norm of vector v induced by the scalar product [·, ·], but one should keep in mind that it doesn’t have p the usual properties of the norm (definiteness and the triangle inequality do not hold), but is used nevertheless due to its relation with the scalar product. 2. The hyperbolic scalar products As mentioned in the introduction, an indefinite scalar product is defined by n n a nonsingular Hermitian indefinite matrix J ∈ C × as [x,y]J = y∗Jx. When J is known from the context, we simply write [x,y] instead of [x,y]J . When J is a signature matrix, i.e., J = diag(±1) := diag(j11,j22,...,jnn), where jkk ∈ {−1, 1} for all k, the scalar product is referred to as hyperbolic and takes the form n [x,y]J = y∗Jx = jiixiyi. i=1 X Throughout this paper we assume that all considered scalar products are hy- perbolic, unless stated otherwise. Indefinite scalar products have another important property which, unfortu- nately, causes a major problem with the construction of the decomposition. A vector v 6=0 is said to be J-degenerate if [v,v] = 0; otherwise, we say that it is J-nondegenerate. Degenerate vectors are sometimes also called J-neutral. If [v,v] < 0 for some vector v, we say that v is J-negative, while we call it J-positive if [v,v] > 0. When J is known from the context, we simply say that the vector is degenerate, nondegenerate, neutral, negative or positive. We extend this notion to matrices as well: a matrix A is J-degenerate if rank A∗JA < rank A. Otherwise, we say that A is J-nondegenerate. Again, if J is known from the context, we simply say that A is degenerate or nondegenerate. 3 We say that the vector v is J-normalized, or just normalized when J is known from the context, if |[v,v]| = 1. As in the Euclidean scalar product, if a vector v is given, then the vector 1 1 v′ = v = v (1) |v| |[v,v]| is a normalization of v. Note that degeneratep vectors cannot be normalized. Also, for a given vector x ∈ Cn, sign[ξx,ξx] is constant for all ξ ∈ C \ {0}. This means that the normalization (1) does not change the sign of the scalar product, i.e., sign[v,v] = sign[v′,v′]=[v′,v′]. Like in the Euclidean scalar products, we define the J-conjugate trans- [ ]J pose (or J-adjoint) of A with respect to a hyperbolic J, denoted as A ∗ , [ ]J n as [Ax,y]J = [x, A ∗ y]J for all vectors x,y ∈ C . It is easy to see that [ ]J [ ] A ∗ = JA∗J. Again, if J is known from the context, we simply write A ∗ . The usual structured matrices are defined naturally. A matrix H is called [ ] J-Hermitian (or J-selfadjoint) if H ∗ = H, i.e., if JH is Hermitian. A matrix [ ] 1 U is said to be J-unitary if U ∗ = U − , i.e., if U ∗JU = J. Like their traditional counterparts, J-unitary matrices are orthonormal with respect to [·, ·]J .
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